PreCalculus
posted by Shreya on .
Rational Functions
does the following function have a hole or a vertical asymptote or both? how do you know? find the yvalue at that point.
y=x^2+7x+12/x^2+8x+15
I factored this out to
y = (x+3)(x+4)/(x+5)(x+3)
The x+3's cancelled out. I don't get the whole verticle and horizontal asymptote stuff, how do u know w/o using technology whether there is an asymptote and if there is a hole?

If a pair of factors cancel, then you would get a hole
in your case, there is a hole when x = 3
If you have a factor in the denominator which does not cancel, it will cause a vertical asymptote
in your case there will be a vertical asymptote at x = 5
BTW, you should use brackets when typing a function like yours
y=(x^2+7x+12)/(x^2+8x+15)
= (x+3)(x+4)/( (x+5)(x+3) ) 
there is a vertical asymptote if the denominator is zero and the numerator is not zero
there is a horizontal asymptote if the degree of the numerator is less than or equal to the degree of the denominator
There is a slant asymptote if the degree of the numerator is 1 more than the degree of the denominator.
I can't believe these facts were not covered in your book.
So,
y = (x+4)/(x+5) as long as x ≠ 3
If x = 3, y = 0/0 which is not defined. There is a "hole" at (3,1/2)
So, you should be able to figure the asymptotes present.
Visit wolframalpha.com and type in
plot (x^2+7x+12)/(x^2+8x+15)
you can play around with lots of stuff there